Combined functionals as risk measures 1 Introduction

Combined functionals as risk measures A.Novosyolov Institute of Computational Modeling SB RAS, Academgorodok, Krasnoyarsk, Russia, 660036 e-mail: [email protected], phone +7 3912 495382 Abstract Risk measures are widely used in insurance pricing, portfolio selection, and in decision-making in general. Two prevalent classes of risk measures are expected utility (a dollar transform), and distorted probability (a probability transform). Both approaches exhibit properties which are not supported by empirical evidence on decision-making under risk. We propose a combined functional (dollar and probability transform) which may combine advantages of both approaches. The present paper develops representation theorems and axiomatic descriptions, presents applications to decision-making under risk, premium calculation, and portfolio selection; and includes numeric and graphical illustrations. Key words: risk measure, expected utility, distorted probability, combined functional, premium calculation, portfolio selection, decision-making.

1

Introduction

Measuring risk continues to be both a hot research topic and an important practical issue. An investor might consider a problem: given a project bringing uncertain gain represented by a random variable X, is it worth investing a certain amount a into the project? Or, taken a bit broader: what is the maximal certain amount a that is worth investing into the project? A property owner faces a similar question: given uncertain losses of property X during a specified period of time, is it worth paying an insurance premium a to transfer the risk X? Or what is the maximal premium a that the owner would be willing to pay for insurance? In both cases the certain amount a may be treated as the certain equivalent of the risk X. This certain equivalent constitutes a functional on the space of risks (random

variables or their distributions), provided that each risk possesses the one. The functional (or its strictly monotone transform) is usually called risk measure. A long used risk measure was EX, the expected value of X. It is actually sometimes used nowadays in life insurance for premium calculation. As long as almost 3 hundred years ago Daniel Bernoulli had pointed out that using expected value leads to the so called St.Petersburg paradox [3], see also [10]; in other words, expected value principle corresponds to a risk-neutral individual, and works poorly for more common risk averse individuals. Thus the expected value turns out to be not an appropriate risk measure. Bernoulli proposed calculation of expected utility of X instead of its expected value; the trick may be treated as dollar transform with utility function U as transforming function. Later the expected utility principle had received a solid foundation in the book by J. von Neumann and O. Morgenstern [7], which was the first axiomatic construction of a risk measure. Due to this foundation expected utility remains the main tool for decision-making. In spite of popularity, expected utility measure has a number of flaws, including linearity with respect to mixture of distributions. Thus it is not surprising that other risk measures were intensively sought. Recently [11] a distorted probability measure was introduced for insurance pricing. A more general class of coherent risk measures had appeared in [2] right after. These happen to be nonlinear in distribution, thus lacking the flaw of expected utility, but unfortunately hold another sort of linearity, specifically, they are positive homogeneous, which is also often undesirable. The present paper is devoted to a combination of expected utility and distorted probability principles. The paper is organized as follows. Section 2 introduces the concepts of risk and risk measure, as well as basic mathematical structures needed in what follows. Section 3 presents methods of defining risk measures via sets of acceptable risks and operators over sets of functionals. Sections 4 and 5 describe expected utility and distorted

probability functionals, and analyze their properties and disadvantages. In section 6 a version of representation theorem for distorted probability measure is stated and proven. Section 7 contains introduction to the combined functional, including representation theorem and illustrations.

2

Risk measures

In this section we will define a concept of risk measure and present its usage in decisionmaking problems. First we need to define the concept of risk. In what follows risk is a random variable which represents uncertain gains (losses if negative) of a decision, or its distribution function. Let (Ω, B, P) be a probability space. Consider a set X of all (almost surely) bounded random variables X : Ω → R, which is usually denoted by X = L∞ (Ω, B, P). Denote I ∈ X the identical unity: I(ω) = 1, ω ∈ Ω. A (cumulative) distribution function FX for a risk X is defined as FX (x) = P(ω ∈ Ω : X(ω) ≤ x), x ∈ R, and its inverse is denoted by FX−1 (v) = sup{x : FX (x) ≤ v}, v ∈ [0, 1]. Expectation of a random variable X (with respect to the probability measure P) is as usual EX =

Z

X(ω) dP(ω) =

Ω

Z

∞

−∞

t dFX (t) =

Z 0

1

FX−1 (v) dv

=−

Z 0

1

FX−1 (v) d(1 − v).

(1)

We will also need expectations with respect to other probability measures Q on the measurable space (Ω, B): EQ X =

Z

X(ω) dQ(ω).

Ω

For the sake of simplicity we will often use a finite sample space: |Ω| = n. In this case a probability measure Q on (Ω, B) is completely defined by an n-tuple Q = (q1 , . . . , qn ) such that q1 ≥ 0, . . . , qn ≥ 0 and q1 + · · · + qn = 1, in particular P = (p1 , . . . , pn ). A random variable X ∈ X is represented by an n-tuple X = (x1 , . . . , xn ), and X may be identified with Rn . Expectation with respect to a generic probability measure Q takes

the form n X

EQ X =

x i qi ,

(2)

i=1

in particular, expectation with respect to the basic probability measure P equals to

EX =

n X

xi pi .

(3)

i=1

Risk measure is any real-valued functional µ : X → R defined on the set of risks X . If a value of a risk measure µ(X) is completely defined by the distribution function FX of X for any X ∈ X , then the risk measure may be thought of as a functional defined on F, the set of all distribution functions with bounded support1 . We will call such risk measures regular2 . Recall two concepts of stochastic dominance here. Let F, G ∈ F be distribution functions. F dominates G in the sense of first stochastic dominance: G ≤1 F , if F (t) ≤ G(t), t ∈ R. Note that for X, Y ∈ X , X ≤ Y implies FX ≤1 FY . Next, for a distribution function F ∈ F consider the integral distribution function

F

(2)

(x) =

Z

x

F (t) dt.

−∞

For F, G ∈ F, F dominates G in the sense of second stochastic dominance3 : G ≤2 F , if F (2) (t) ≤ G(2) (t), t ∈ R. Note that first and second stochastic dominance constitute partial orderings on X (or F). In what follows we will call a risk measure µ : X → R monotone with respect to partial order ≤, if X ≤ Y implies µ(X) ≤ µ(Y ). The similar concept applies to risk measures of the form µ : F → R. 1

We say that a distribution function F has a bounded support [a, b] if −∞ < a < b ≤ ∞, F (x) = 0 for x < a and F (x) = 1 for x ≥ b. 2 In [6] such risk measures were called law invariant. 3 Remind that we treat X as gain. If X is treated as loss, using integral of decumulative distribution function SX (t) = P(X > t), t ∈ R in the definition of the second stochastic dominance is more appropriate.

3

Defining risk measures

Risk measure, as a functional on X or F, may be defined explicitly or by describing its properties. In either case the result might be a class of functionals, possessing specified properties (axioms) or derived from explicit formula. In the present section we present two ways of axiomatic definition of risk measures. First relies on a concept of acceptable risks, while second uses a family of functionals to produce a new functional. Now let us define some properties of risk measures for future using. A risk measure µ : X → R is called translation invariant if µ(X + aI) = µ(X) + a, X ∈ X , a ∈ R.

(4)

µ is called monotone in value if X ≤ Y =⇒ µ(X) ≤ µ(Y ), X, Y ∈ X .

(5)

µ is called sub(super)additive if µ(X + Y ) ≤ (≥)µ(X) + µ(Y ), X, Y ∈ X .

(6)

µ is called positively homogeneous if µ(λX) = λµ(X), X ∈ X , λ ≥ 0.

(7)

Denote L+ ⊆ X the nonnegative cone of X , that is, the set of all random variables X ∈ X taking nonnegative values with probability 1: L+ = {X ∈ X : P(ω ∈ Ω : X(ω) ≥ 0) = 1} and L− = −L+ – non-positive cone. Denote A ⊂ X a set of risks that we would treat as acceptable, that is, any risk X ∈ A may be taken without additional reward. The following assumptions about this set L+ ⊆ A,

L− ∩ A = {0}

(8)

seem reasonable. The first inclusion in (8) states that any nonnegative risk is acceptable. The second one states that only constant zero risk is acceptable among non-positive risks: if a risk does not bring gain in any state of nature, and brings loss in some states, it cannot be accepted without additional reward. Now, given any acceptance set A, possessing properties (8), one can define the corresponding risk measure µA : X → R as follows: µA (X) = sup{r ∈ R : X − rI ∈ A}.

(9)

If X is acceptable, then µA (X) means the maximal certain amount (capital), that may be subtracted from X without leaving acceptance set. If X is not acceptable (X 6∈ A), then µA (X) is non-positive, and its absolute value indicates how large certain capital should be added to X for moving X into acceptance set. In either case µA (X) may be treated as certain equivalent for X. Note that conditions (8) ensure that µA in (9) takes only finite values. Note also that any risk measure defined via (9) is translation invariant and monotone. Given any monotone, translation invariant, finite risk measure µ : X → R, the corresponding acceptance set A = Aµ is defined as Aµ = {X ∈ X : µ(X) ≥ 0}.

(10)

Next consider building risk measures using a class of predefined functionals. Let Λ be an index set and M be a set of functionals: M = {µλ , λ ∈ Λ}, so that µλ : X → R for each λ ∈ Λ. Then extremal functionals µsΛ (X) = sup µλ (X), X ∈ X

(11)

µiΛ (X) = inf µλ (X), X ∈ X

(12)

λ∈Λ

and λ∈Λ

constitute new risk measures on X . Further, if Λ is endowed with a probability space structure (Λ, C, S), then averaging with respect to the probability measure S also provides a new risk measure µaΛ (X)

=

Z Λ

µλ (X) dS(λ), X ∈ X .

(13)

Extremal operation are often used in conjunction with a class of expectations as basic functionals. In this case each λ ∈ Λ represents a probability measure on (Ω, B), and µλ (X) = Eλ X, X ∈ X , λ ∈ Λ.

(14)

An example of using (12), (14) is given below in representation theorem for coherent risk measures in section 5. An example of using averaging operation (13) for building distorted probability measure from the class of all VaR measures is presented in section 5. To complete the section, let us state a proposition on preserving risk measures properties under extremal and averaging operations; the proof of the proposition is straightforward. Proposition 3.1 If each µλ , λ ∈ Λ is translation invariant (monotone, positively homogeneous), then µsΛ , µiΛ and µaΛ are also translation invariant (monotone, positively homogeneous). If each µλ , λ ∈ Λ is sub(super)additive then µaΛ and µsΛ (µiΛ ) are also sub(super)additive.

4

Expected utility measure

Expected utility is being used at least since Daniel Bernoulli had constructed his famous St. Petersburg paradox, explaining the flaws of expectation as a certainty equivalent for a risky project. Later the expected utility principle was provided with a solid foundation by John von Neumann and Oscar Morgenstern in their seminal book [7]. The expected utility functional ρ : X → R is defined via distribution function of an argument, thus

being regular: ρ(X) = ρU (X) = EU (X) =

Z

∞

−∞

U (t) dFX (t) =

Z

1

0

U (FX−1 (v)) dv.

(15)

Here U : R → R stands for utility function, a parameter of the functional (15). Denote U0 (x) = x, x ∈ R. Clearly, ρU0 coincides with expectation functional ρU0 (X) = EX, X ∈ X , so expectation is a special case of (15). In general, expected utility functional may be treated as calculating expectation of dollar transform. Commonly used classes of utility functions are exponential U (t) = (1 − exp(−αt))/(1 − exp(−α)), α > 0, t ∈ R,

(16)

U (t) = tα , t ≥ 0, α ∈ (0, 1)

(17)

U (t) = ln(1 + αt)/ ln(1 + α), t > −1/α, α > 0.

(18)

power

and logarithmic

The last two classes are intended for risks bounded from below, e.g. nonnegative risks. Taking limit as α → 0 in (16) and (18), and as α → 1 in (17), provides boundary utility U0 . When parameter α satisfies conditions given in (16) – (18), all these utility functions are increasing and concave, and corresponding expected utility functionals are monotone with respect to first and second stochastic dominance. The main disadvantage of expected utility functional is its linearity with respect to mixture of distributions; given F, G ∈ F such that µ(F ) = µ(G), the following equality holds: µ(δF + (1 − δ)G) = µ(F ) = µ(G), δ ∈ [0, 1].

(19)

This means that if two risks F, G are equivalent (in the sense of preference), then their convex combinations are also equivalent to both F and G. As experiments show [4], this

feature often lacks from real human preferences. Thus expected utility functional may be regarded as a linear approximation (perhaps, very poor) to a decision-maker’s preferences.

5

Distorted probability measure

Distorted probability measure was introduced in [11] for calculation of insurance premium for nonnegative risks (losses) and was generalized in [12] to risks taking both positive and negative values: πg (X) =

Z

0

−∞

[g(1 − FX (t)) − 1] dt +

Z 0

∞

g(1 − FX (t)) dt.

(20)

Here g : [0, 1] → [0, 1] is a distortion function, which is nondecreasing and satisfies g(0) = 0, g(1) = 1. A simple algebra provides a representation πg (X) = −

1

Z 0

FX−1 (v) dg(1 − v), X ∈ X ,

(21)

which resembles the last expression for expectation in (1). This allows treating πg as expectation calculated with probability transform. Note that in case g0 (v) = v, v ∈ [0, 1] the functional πg0 coincides with expectation: πg0 (X) = EX, X ∈ X . Note also that the functional is regular, so it may be considered as a functional on F: πg (F ) = −

Z

1

F −1 (v) dg(1 − v), F ∈ F.

(22)

0

It is clear from (20) that since g is nondecreasing, the functional πg is monotone in value and with respect to the first stochastic dominance. It is also straightforward that πg is translation invariant and positively homogeneous. Monotonicity with respect to the second stochastic dominance requires additional assumptions. Proposition 5.1 Distorted probability functional πg is monotone with respect to the second stochastic dominance iff the distortion function g is convex.

Moreover, convexity of g provides more properties of πg , stated in the following proposition Proposition 5.2 Let g be a convex function. Then the functional πg is super-additive, concave in value: πg (δX + (1 − δ)Y ) ≥ δπg (X) + (1 − δ)πg (Y )), X, Y ∈ X , δ ∈ [0, 1]

(23)

and convex in distribution: πg (δF + (1 − δ)G) ≤ δπg (F ) + (1 − δ)πg (G), F, G ∈ F, δ ∈ [0, 1].

(24)

Proofs of propositions 5.1 and 5.2 may be found in [8]. Artzner et al [2] had recently introduced a class of risk measures generalizing a distorted probability measure, which they called coherent risk measures. Here we will briefly describe some properties of coherent risk measures that are necessary for what follows. Our definition of coherent risk measures slightly differs from that of [2]. Definition 5.1 A risk measure µ : X → R is called coherent, if it is translation invariant, monotone in value, positive homogeneous and superadditive4 . The following representation theorem is a reformulation of the proposition 4.1 from [2] for the current setting. Theorem 5.1 A risk measure µ : X → R is coherent iff there exists a set of probability measures Q on (Ω, B) such that µ(X) = µQ (X) = inf EQ X, X ∈ X . Q∈Q

(25)

We will call any set of measures Q, satisfying (25) a generator for the risk measure µQ . 4

In [2] the risk measure −µ is studied, which is sub-additive.

Remark 5.1 It can be easily seen that given a set of probability measures Q, representing e in between: Q ⊆ Q e ⊆ the risk measure µQ , the closed convex hull5 Co(Q) and any set Q

Co(Q) generate the same risk measure µQ = µQe = µCo(Q) by (25). This observation allows choosing different generating sets of measures for a risk measure as appropriate. As was mentioned before, the distorted probability measure is positively homogeneous, monotone in value and translation invariant. Proposition 5.2 ensures that the functional is super-additive, provided that distortion g is convex. This leads to the Proposition 5.3 Distorted probability measure πg is coherent, provided that the distortion function g is convex. Now consider some examples of convex distortion functions. Power family is given by g(v) = v 1/β , v ∈ [0, 1], β ∈ (0, 1].

(26)

Dual power functions have the form g(v) = 1 − (1 − v)β , v ∈ [0, 1], β ∈ (0, 1].

(27)

g(v) = (exp(βv) − 1)/(exp(β) − 1), v ∈ [0, 1], β ∈ (0, 1].

(28)

Exponential family:

Limiting cases β = 1 in (26), (27) and β → 0 in (28) represent pure expectation πg0 (X) = EX, X ∈ X . More examples may be found in [11]. Concave distortion functions ge, presented therein, may be converted to dual convex functions by the transform g(v) = 1− ge(1−v), v ∈ [0, 1]. 5

Closed convex hull Co(A) of a set A consists of all finite convex combinations of elements of A, that

is, Co(A) =

( k X i=1

λi ai , ai ∈ A, λi ≥ 0, i = 1, . . . , k,

k X i=1

) λi = 1, k = 1, 2, . . .

Coherent risk measures as well as their special case, distorted probability measures, are positive homogeneous. This property is often undesirable. Consider a lottery paying a moderate amount a in case of win, that occurs with probability p (and paying nothing with probability 1 − p). Let the price of the lottery be b. A risk averse person willing to pay the specified price for this lottery might think, that the price 1000b for a similar lottery with winning payment 1000a, is too large. On the other hand, consider an insurance portfolio consisting of a number of risks, each bringing loss −a with probability p, and insurance premium b. If a single similar risk with loss size −1000a is to be added to the portfolio, insurance company would hardly consider a premium size 1000b as sufficient. We can now conclude that both expected utility and distorted probability measures possess a sort of linearity, which prevents from proper reflection of individual risk aversion. In section 7 we will propose a combined functional, which allows getting rid of both sorts of linearity, thus enabling a more flexible environment for reflecting individual preferences. The next section will be devoted to deriving a specific form of the representation theorem 5.1 for distorted probability measure.

6

Representation theorem

Let us consider the representation theorem 5.1 for the special case of distorted probability measure in more detail. This consideration contains essential part of corresponding theorem for combined risk measures as well. We would confine ourselves to the case of finite sample space |Ω| = n to restrict technicalities. Remind that in this case X = Rn , any risk X ∈ X is represented by an n-tuple (x1 , . . . , xn ), expectation with respect to the basic probability measure P = (p1 , . . . , pn ) and generic probability measure Q = (q1 , . . . , qn ) are expressed as in (3) and (2).

Let us fix a convex distortion function g, then the theorem 5.1 guarantees existing of a set of probability measures Q such that6 πg (X) = min EQ X. Q∈Q

Thanks to remark 5.1, Q may be assumed closed and convex to avoid ambiguity. From (21) in case x1 ≤ · · · ≤ xn one easily obtains πg (X) =

n X

x k qk ,

(29)

k=1

where qk = g(rk ) − g(rk+1 ), rk =

n X

pi , k = 1, . . . , n, rn+1 = 0.

i=k

It is clear that (29) equals to the expectation EQ X with respect to probability measure Q = (q1 , . . . , qn ). If components of X are not sorted in ascending order, the expression (29) remains valid with different probability measure Q. To calculate the latter consider the set Γ of all n! permutations of the set {1, . . . , n}. For a given X ∈ X let γ = γX ∈ Γ be a permutation, for which xγ(1) ≤ xγ(2) ≤ . . . ≤ xγ(n) . Denoting γ γ ), rkγ = qγ(k) = g(rkγ ) − g(rk+1

n X

γ = 0, pγ(k) , k = 1, . . . , n, rn+1

(30)

i=k

one concludes that (29) turns to πg (X) =

n X

xk qkγ

(31)

k=1

with components of Qγ = (q1γ , . . . , qnγ ) calculated via (30). Thus πg (X) = EQγ X with γ = γX does not exceed expectations of X with respect to measures Qγ , corresponding to other permutations γ ∈ Γ. Since components of any X ∈ X may be sorted in ascending order by a permutation γ ∈ Γ, there is a set of at most n! probability measures Q0 = {Qγ , γ ∈ Γ} such that 6

In finite-dimensional case inf may be substituted with min without trouble.

πg (X) = EQ X for some Q ∈ Q0 and πg (X) = min EQ X. Q∈Q0

Noting that Q = Co(Q0 ), we conclude the proof of the following Theorem 6.1 Let g be a convex distortion function, |Ω| = n, and Γ be the set of all permutations of {1, . . . , n}. The distorted probability measure πg is generated by the set of probability measures Q0 = {Qγ , γ ∈ Γ} with components of Qγ defined by (30). Equivalently, the generator of πg is the polyhedron Co(Q0 ). Consider an example. Let |Ω| = 3, P = (1/4, 1/4, 1/2), and g(v) = v 2 , v ∈ [0, 1]. The set of all probability measures constitute the standard simplex in R3 . Table 1 lists all permutations γ ∈ Γ and corresponding probability measures Qγ , which are vertices of the polyhedron Co(Q0 ). Figure 3 presents projections of the generators Co(Q0 ) of πg onto the plane of standard simplex for the specified power distortion and dual power distortion g(v) = 1 − (1 − v)0.5 . γ (1,2,3) (1,3,2) (2,1,3) (2,3,1) (3,1,2) (3,2,1)

Qγ (7/16,5/16,1/4) (7/16,1/16,1/2) (5/16,7/16,1/4) (1/16,3/4,3/16) (1/16,7/16,1/2) (1/16,3/16,3/4)

EQγ X 44/16 36/16 52/16 70/16 60/16 52/16

Table 1: Generator Q0 and expectation for X = (1, 5, 3)

7

Combined risk measure

Since main flaws of expected utility (dollar transform) and distorted probability (probability transform) risk measures are due to linearity, a natural way of overcoming disadvantages would be simultaneous usage of both transforms. The thought leads to a class

of combined functionals µU,g (X) = −

Z 0

1

U (FX−1 (v)) dg(1 − v),

(32)

arising from (1) by applying dollar transform to the integrand and probability transform to the differential part. Here U stands for an utility function as described in section 4, and g stands for distortion function, which was introduced in section 5. This two-parameter class provides much flexibility; it clearly contains all expected utility measures (when g(v) ≡ v) and all distorted probability measures (when U (t) ≡ t). If U is concave and g is convex, then the functional (32) is concave in value, and may be used in a wide range of decision-making applications for risk averse individuals. Most of the analysis, presented in sections 4 through 6 directly applies to the combined functional. This risk measure is not positive homogeneous, thus it is not coherent. It is concave in the sense similar to that of [5], and allows similar axiomatic introduction. In contrast to convex risk measures of [5], the combined risk measure is regular. It is also monotone with respect to first and second stochastic dominance, provided that parameter functions possess the same necessary properties, as in partial cases (15) and (20). Here we will state a representation theorem for (32) and illustrate properties of the functional by its indifference curves, which are shown on figure 4. Parameters in this illustration were adjusted to achieve high similarity of indifference curves with quite different parameter functions to emphasize flexibility of the functional. Theorem 7.1 Let |Ω| = n, the functional (32) be given with concave nondecreasing utility function U and convex nondecreasing distortion function g. Then µU,g (X) = min EQ U (X), Q∈Q0

where Q0 is the same as in theorem 6.1. Proof. Since U is a monotone function, the proof essentially repeats that of theorem 6.1.

8

Conclusion

The paper provides an overview of some methods of building risk measures, which represent human preferences over risky projects. Special attention is paid to expected utility and distorted probability risk measures, that may be treated as applying dollar transform and probability transform, respectively, before calculation of expectation. Since both approaches exhibit a sort of linearity, the combined functional is proposed and studied to some extent. The main result of the paper is the representation theorem for distorted probability and combined risk measures. Interesting directions of further research include representing combined risk measures in terms of acceptance sets as in section 3, and in terms of individual preference relation over risks, as was implemented in [8] for some general risk measures. Inverse problems deserve special attention; since greater flexibility of combined risk measures may cause more trouble in selecting the best measure from the class. A few illustrations are presented directly in the paper. Much more may be found in the presentation and the executable, accompanying the paper on the CAS site, and also available from the author’s site. The author would like to express many thanks to Louis Anthony Cox for multiple discussions on expected utility and other decision-making topics.

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[11] Wang, S.(1996) Premium calculation by transforming the layer premium density. ASTIN Bulletin, 26, pp. 71-92. [12] Young V.R. (1999) Discussion of Christofides’ Conjecture Regarding Wang’s Premium Principle. ASTIN Bulletin, 29, 2, 191–195.

a) power, α = 0.6 b) exponential, α = 0.4 c) logarithmic, α = 0.9 Figure 1. Expected utility indifference cirves

a) power, β = 0.5 b) dual power, β = 0.5 c) exponential, β = 1.5 Figure 2. Distorted probability indifference cirves

a) power, β = 0.5

b) dual power, β = 0.5 Figure 3. Generator for distorted probability

a) logarithmic utility, α = 0.9; power distortion, β = 0.7

b) exponential utility, α = 0.2; exponential distortion, β = 1.2 Figure 4. Combined functional indifference cirves